I just discovered it and this is my first time applying for something. The deadline is Feb 12 and it was posted sept. 8, 2014. Will it hurt my chances that I will turn in the deadline super close to the due date? Or should I wait for next year, and turn in the application early? Also are these internships super selective? Thanks

I would not count on its being a valuable credential on your resume. It would translate an interest in medicine, but you could do that another cheaper way. These expensive summer programs are not very competitive and are largely fundraisers for the college. I would suggest looking into competitive unpaid internships at hospitals. Even shadowing a doctor for a week or two would be a worthwhile experience.

I have another question. I am specifically looking into doing cardiothoracic surgery, would it be better to pursue an internship at hospital that looks into a variety of medical programs that is competitive, or do this one, which specializes in my field that I want to go into?

At this point you are not yet an undergraduate, right? If you want to become a doctor some kind of summer internship in the medical field makes sense but it's too early to worry about specialty within the medical field

Also, I was looking at this internship, also very expensive.http://www.projects-abroad.org/doctors-without-borders-alternative/
Is it the same as a Stanford internship, not competitive like OGP student said, and not be a good credential for my college resume? Or would it be worth while, and improve my chances of getting into Stanford or John Hopkins University? Thanks